In a previous notebook, I have shown properties of the distribution of stars in the sky. Here, I would like to use the existing database of stars' positions and display them as a triangulation

Let's first initialize the notebook:

In [1]:

import numpy as np
np.set_printoptions(precision=6, suppress=True)
%matplotlib inline
%config InlineBackend.figure_format = 'svg'
import matplotlib.pyplot as plt
phi = (np.sqrt(5)+1)/2
fig_width = 10
figsize = (fig_width, fig_width/phi)

importing data¶

Importing all stars' position is as simple as invoking the HYG database:

In [2]:

import pandas as pd
url = "https://github.com/astronexus/HYG-Database/raw/master/hygdata_v3.csv"
space = pd.read_csv(url, index_col=0)

In [3]:

print(f'Columns in the database: {space.columns=}')

Columns in the database: space.columns=Index(['hip', 'hd', 'hr', 'gl', 'bf', 'proper', 'ra', 'dec', 'dist', 'pmra',
       'pmdec', 'rv', 'mag', 'absmag', 'spect', 'ci', 'x', 'y', 'z', 'vx',
       'vy', 'vz', 'rarad', 'decrad', 'pmrarad', 'pmdecrad', 'bayer', 'flam',
       'con', 'comp', 'comp_primary', 'base', 'lum', 'var', 'var_min',
       'var_max'],
      dtype='object')

In [4]:

print(f'Number of stars in the catalog = {len(space)=}')

Number of stars in the catalog = len(space)=119614

extracting ra, dec and mag¶

For which we may extract what interests us: position (right ascension and declination) and visual magnitude:

In [5]:

space_pos = space[['ra', 'dec', 'mag']]
space_pos

Out[5]:

	ra	dec	mag
id
0	0.000000	0.000000	-26.70
1	0.000060	1.089009	9.10
2	0.000283	-19.498840	9.27
3	0.000335	38.859279	6.61
4	0.000569	-51.893546	8.06
...	...	...	...
119611	23.963895	38.629391	12.64
119612	23.996567	47.762093	16.10
119613	23.996218	-44.067905	12.82
119614	23.997386	-34.111986	12.80
119615	0.036059	-43.165974	13.05

119614 rows × 3 columns

First, right ascension is "the angular distance of a particular point measured eastward along the celestial equator from the Sun at the March equinox to the (hour circle of the) point in question above the earth." It is given in hours:

In [6]:

ra_min, ra_max = 0, 24
print(f"RA: {space_pos['ra'].min()=}, {space_pos['ra'].max()=}")

RA: space_pos['ra'].min()=0.0, space_pos['ra'].max()=23.998594

normalizing ra into az¶

Let's convert this in visual angle, that is in an azimuth:

In [7]:

space_norm = space_pos[['dec', 'mag']].copy()
space_norm

Out[7]:

	dec	mag
id
0	0.000000	-26.70
1	1.089009	9.10
2	-19.498840	9.27
3	38.859279	6.61
4	-51.893546	8.06
...	...	...
119611	38.629391	12.64
119612	47.762093	16.10
119613	-44.067905	12.82
119614	-34.111986	12.80
119615	-43.165974	13.05

119614 rows × 2 columns

In [8]:

az_min, az_max = 0, 360
def ra2az(ra):
    return az_max - ra / ra_max * az_max

In [9]:

space_norm["az"] =  ra2az(space_pos['ra'])

In [10]:

print(f"AZ: {space_norm['az'].min()=}, {space_norm['az'].max()=}")

AZ: space_norm['az'].min()=0.021090000000015152, space_norm['az'].max()=360.0

Then, declination is "comparable to geographic latitude, projected onto the celestial sphere" and is given here in degrees:

In [11]:

dec_min, dec_max = -90, 90
print(f"DEC: {space_norm['dec'].min()=}, {space_norm['dec'].max()=}")

DEC: space_norm['dec'].min()=-89.782428, space_norm['dec'].max()=89.569427

The magnitude varies a lot (the less the value, the more it is visible - "The apparent magnitudes of known objects range from the Sun at −26.7 to objects in deep Hubble Space Telescope images of magnitude +31.5"):

In [12]:

print(f"mag: {space_norm['mag'].min()=}, {space_norm['mag'].max()=}")

mag: space_norm['mag'].min()=-26.7, space_norm['mag'].max()=21.0

Let's normalize the lower bound:

In [13]:

space_norm['mag'] = space_pos['mag'] - space_pos['mag'].min()
print(f"mag: {space_norm['mag'].min()=}, {space_norm['mag'].max()=}")

mag: space_norm['mag'].min()=0.0, space_norm['mag'].max()=47.7

Let's define a threshold using the quantile function:

In [14]:

print(f"{space_norm['mag'].quantile(q=0.01)=}")

space_norm['mag'].quantile(q=0.01)=31.45

Which leaves only a limited number of stars

In [15]:

space_bright = space_norm[space_norm['mag']<space_norm['mag'].quantile(q=0.05)]

In [16]:

#space_bright = space_pos[space_pos['mag']<6]

In [17]:

print(f'Number of bright stars = {len(space_bright)=}')

Number of bright stars = len(space_bright)=5955

In [18]:

print(f"MAG: {space_bright['mag'].min()=}, {space_bright['mag'].max()=}")

MAG: space_bright['mag'].min()=0.0, space_bright['mag'].max()=32.85

scatter plots¶

From these elements, we may plot the stars on these coordinates:

In [19]:

stars_color = np.array([255., 235., 220.])
stars_color /= 255.

In [35]:

fig, ax = plt.subplots(figsize=(fig_width, fig_width/phi))
ax.set_facecolor('black')
ax.scatter(space_bright['az'], space_bright['dec'], c=stars_color[None, :], ec='none', s=.1 * (space_bright['mag'].max()-space_bright['mag']))
ax.set_xlabel('Azimuth')
ax.set_ylabel('Declination')
ax.set_xlim(az_min, az_max)
ax.set_ylim(dec_min, dec_max);

do you see the Milky way?

Orion¶

We may want to focus on the Orion constellation):

In [21]:

orion_az_max, orion_az_min = ra2az(4.8), ra2az(6.25)
orion_dec_min, orion_dec_max = -13.0, 22.
space_orion = space_bright[(orion_az_min < space_bright['az']) * (space_bright['az'] < orion_az_max) *
                                (orion_dec_min < space_bright['dec']) * (space_bright['dec'] < orion_dec_max)]

In [22]:

print(f'Number of bright stars in orion = {len(space_orion)=}')

Number of bright stars in orion = len(space_orion)=192

In [23]:

print(f"MAG: {space_orion['mag'].min()=}, {space_orion['mag'].max()=}")

MAG: space_orion['mag'].min()=26.88, space_orion['mag'].max()=32.85

In [24]:

fig, ax = plt.subplots(figsize=(fig_width, fig_width))
ax.set_facecolor('black')
ax.scatter(space_orion['az'], space_orion['dec'], c=stars_color[None, :], s= 10 * (space_orion['mag'].max()-space_orion['mag']))
ax.set_aspect('equal')
ax.set_xlabel('Azimuth')
ax.set_ylabel('Declination')
ax.set_xlim(orion_az_min, orion_az_max)
ax.set_ylim(orion_dec_min, orion_dec_max);

Which is to be compared to this image:

In [25]:

from  IPython.display import display, SVG, Image
display(SVG('https://upload.wikimedia.org/wikipedia/commons/f/ff/Orion_IAU.svg'))

Triangulation¶

Following work with Etienne Rey, we have developped visualization based on triangulation of an optimal packing of point clouds.

In [26]:

display(Image('https://laurentperrinet.github.io/post/2021-10-04_interstices/featured.jpg'))

The idea here is to apply such a visualization to the stars position. IT is based on

In [27]:

import matplotlib.tri as tri
triang = tri.Triangulation(space_bright['az'], space_bright['dec'])

In [28]:

fig, ax = plt.subplots(figsize=(fig_width, fig_width))
ax.set_facecolor('black')
ax.triplot(triang, lw=0.1, color=stars_color[None, :])
ax.set_aspect('equal')
ax.set_xlabel('Azimuth')
ax.set_ylabel('Declination')
border_ratio = .95
ax.set_xlim(az_min+(az_max-az_min)*(1-border_ratio), az_min+(az_max-az_min)*border_ratio)
ax.set_ylim(dec_min+(dec_max-dec_min)*(1-border_ratio), dec_min+(dec_max-dec_min)*border_ratio);

In [29]:

fig, ax = plt.subplots(figsize=(fig_width, fig_width))
ax.set_facecolor('black')
ax.triplot(triang, lw=0.3, c=stars_color)
ax.scatter(space_orion['az'], space_orion['dec'], c=stars_color[None, :], ec='none', alpha=.5, s= 40 * (space_orion['mag'].max()-space_orion['mag']))
ax.set_aspect('equal')
ax.set_xlabel('Azimuth')
ax.set_ylabel('Declination')
ax.set_xlim(orion_az_min+(orion_az_max-orion_az_min)*(1-border_ratio), orion_az_min+(orion_az_max-orion_az_min)*border_ratio)
ax.set_ylim(orion_dec_min+(orion_dec_max-orion_dec_min)*(1-border_ratio), orion_dec_min+(orion_dec_max-orion_dec_min)*border_ratio);

In [30]:

q = 0.01
space_norm['mag'].quantile(q=q)

Out[30]:

31.45

In [31]:

space_orion = space_norm[space_norm['mag']<space_norm['mag'].quantile(q=q)]

In [32]:

space_orion = space_orion[(orion_az_min < space_orion['az']) * (space_orion['az'] < orion_az_max) *
                           (orion_dec_min < space_orion['dec']) * (space_orion['dec'] < orion_dec_max)]

Let's now super-impose different triangulations at different magnitudes:

In [33]:

fig, ax = plt.subplots(figsize=(fig_width, fig_width))
ax.set_facecolor('black')

for mag in range(2, 8):
    #q = 1 - 0.8**mag
    #print (q)
    space_orion = space_norm.copy()
    #space_orion = space_norm[space_norm['mag']<mag]
    space_orion = space_orion[(orion_az_min < space_orion['az']) * (space_orion['az'] < orion_az_max) *
                               (orion_dec_min < space_orion['dec']) * (space_orion['dec'] < orion_dec_max)]
    space_orion = space_orion[space_orion['mag'] > space_orion['mag'].max()-mag]
    print(f'{mag=:.3f}, Number of bright stars = {len(space_orion)=}')
    triang = tri.Triangulation(space_orion['az'], space_orion['dec'])
    ax.triplot(triang, lw=9/mag**2, color=stars_color, alpha=.4)
#ax.scatter(space_orion['az'], space_orion['dec'], s= 40 * (space_orion['mag'].max()-space_orion['mag']))
ax.set_aspect('equal')
ax.set_xlabel('Azimuth')
ax.set_ylabel('Declination')
ax.set_xlim(orion_az_min+(orion_az_max-orion_az_min)*(1-border_ratio), orion_az_min+(orion_az_max-orion_az_min)*border_ratio)
ax.set_ylim(orion_dec_min+(orion_dec_max-orion_dec_min)*(1-border_ratio), orion_dec_min+(orion_dec_max-orion_dec_min)*border_ratio);

mag=2.000, Number of bright stars = len(space_orion)=8
mag=3.000, Number of bright stars = len(space_orion)=14
mag=4.000, Number of bright stars = len(space_orion)=25
mag=5.000, Number of bright stars = len(space_orion)=51
mag=6.000, Number of bright stars = len(space_orion)=140
mag=7.000, Number of bright stars = len(space_orion)=452

some book keeping for the notebook¶

In [34]:

%load_ext watermark
%watermark -i -h -m -v -p numpy,matplotlib,scipy,pillow,imageio  -r -g -b

Python implementation: CPython
Python version       : 3.9.9
IPython version      : 7.30.0

numpy     : 1.20.3
matplotlib: 3.4.3
scipy     : 1.6.0
pillow    : not installed
imageio   : 2.13.1

Compiler    : Clang 13.0.0 (clang-1300.0.29.3)
OS          : Darwin
Release     : 21.1.0
Machine     : x86_64
Processor   : i386
CPU cores   : 4
Architecture: 64bit

Hostname: ekla

Git hash: 6b0c86af026462cfae61f02aef25a3acb568faf9

Git repo: https://github.com/laurentperrinet/sciblog.git

Git branch: master